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Note on total and partial functions in second-order arithmetic Title (Proof Theory, Computation Theory and Related Topics)

Author(s) 藤原, 誠; 佐藤, 隆

Citation 数理解析研究所講究録 (2015), 1950: 93-97

Issue Date 2015-06

URL http://hdl.handle.net/2433/223934

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University 数理解析研究所講究録 第 1950 巻 2015 年 93-97 93

Note on total and partial functions in second-order arithmetic

Makoto Fujiwara* Takashi Sato Mathematical Institute Mathematical Institute Tohoku University Tohoku University 6-3, Aramaki Aoba, Aoba-ku Sendai, Miyagi, Japan [email protected]

Abstract We analyze the total and partial extendability for partial functions in second-order arith- metic. In particular, we show that the partial extendability for consistent of finite partial functions and that for consistent of partial functions are equivalent to WKL (weak K\"onig’s lemma) over $RCA_{0}$ , despite the fact that their total extendability derives ACA (arith- metical comprehension).

1 Introduction

The mainstream of reverse has been developed in second-order arithmetic, whose language is set-based and functions are represented as their graphs [4]. For this reason, some results on partial functions in reverse mathematics do not correspond to those in while many results in computability theory can be transformed into the corresponding results in reverse mathematics straightforwardly. For example, it is known in computability theory

that every computable partial $\{0$ , 1 $\}$ -valued has a total extension with low degree, and also that there exists a computable partial $\{0$ , 1 $\}$ -valued function which has no computable total extension (cf. [1, Lemma 8.17]). One can, however, easily see that the assertion that every $RCA partial $\{0$ , 1 $\}$ -valued function has a total extension is provable in _{0}$ . On the other hand, as we will see in Section 2, the assertion that every partial function has a total extension is equivalent to arithmetical comprehension over $RCA_{0}$ . That is to say, the treatment of partial functions in second-order arithmetic sometimes causes a different situation from computability theory. Based on this insight, we investigate some basic properties on partial functions in the context of reverse mathematics. We recall that in our setting, a function $f$ is a set of pairs such that if $(n,m)$ , $(n,m’)\in$

$n$ $m$ $(n,m)\in f$ $f$ then $m=m’$ . A function $f$ is said to be total if for all there exists such that and to be partial otherwise. We use a notation $f$ : $Xarrow Y$ only for total functions and $f$ $:\ Xarrow Y$ $X$ $Y$ denotes that $f$ is (a graph of) $a$ (possibly partial) function from to . For $f:\subset Xarrow Y,$ $dom(f)$ denotes the domain of $f$ . We refer the reader to Simpson’s book [4] for other basic definitions and comprehensive treatment of ordinary mathematics in second-order arithmetic as well as the coding of pairs, finite sets, finite etc. *The 1st author has been supported by Grant-in-Aid for JSPS Fellows. 94

2 Totalization of Partial Function

We first show that totalization of a partial function requires arithmetical comprehension scheme.

Theorem 1. The following are equivalent over $RCA_{0}.$ 1. ACA.

2. Every partial function $f$ $:\subset \mathbb{N}arrow \mathbb{N}$ has its domain $dom(f)$ , i.e., a set $X\subseteq \mathbb{N}$ such that $\forall n(n\in Xrightarrow\exists m(f(n)=m$

3. Every partialfunction $f:\subset \mathbb{N}arrow \mathbb{N}$ has its total extension $\tilde{f}:\mathbb{N}arrow \mathbb{N}$, i. e., a totalfunction $\tilde{f}:\mathbb{N}arrow \mathbb{N}$ such that $\forall n\forall m(f(n)=marrow\tilde{f}(n)=m)$ . Proof We reason in $RCA_{0}.$ $(1arrow 2)$ is trivial since $dom(f)$ is $\Sigma_{1}^{0}$ definable in $f.$

$(2arrow 3)$ is shown by taking the total extension $\tilde{f}$, by $\Sigma_{0}^{0}$ comprehension, as

$\tilde{f}:=\{(n,m)$ : $(n,m)\in f\vee(n\not\in dom(f)\wedge m=0$

$(3arrow 1)$ is shown via [4, Lemma $m.$ ] $.3$ ] as follows. Let $g:\mathbb{N}arrow \mathbb{N}$ be a one-to-one function.

Take a partial $f$ of $g$ as $f:=\{(n,m):g(m)=n\}$ by $\Sigma_{0}^{0}$ comprehension. By our assumption 3, there exists a total extension $\tilde{f}$ of $f$ . Then it is easy to see that

$\exists n(g(n)=m)rightarrow g(\tilde{f}(m))=m)$ holds for all $m$ . Thus we have the image of $g$ by $\Sigma_{0}^{0}$ comprehension. $\square$ For further understanding, let us consider computable partial functions as their program codes in second-order arithmetic. Note that one can construct the program code for a given (graph of) computable partial function in $RCA_{0}$ , while it seems to be impossible to construct the corre- sponding graph for a given program code of computable partial function in $RCA_{0}$ . This indicates that for a computable partial function, its graph has more information than its program code. The previous theorem states that even if partial functions are given as their graphs, the totalization requires arithmetical comprehension. On the other hand, one can easily see that $RCA_{0}$ proves that every graph of partial $\{0$ , 1 $\}$ -valued function has a total extension in contrast to [1, Lemma 8.17] in computability theory. If partial $\{0$ , 1 $\}$ -valued functions are represented as their program codes, the corresponding fact to [1, Lemma 8.17] holds in second-order arithmetic:

Proposition 2. Thefollowing assertion is provable in $WKL_{0}$ but is not provable in RC $A^{}$ : for any program code $e$ of computable partial $\{0$ , 1 $\}$ -valuedfunction $(i.e., \{e\}:\subset Narrow 2)$, there exists its total extension $f_{e}:\mathbb{N}arrow 2$, i.e., $T(e, i,z)arrow f_{e}(i)=U(z)$ where $T(e, i,z)$ expresses that the with G\"odel number $e$ applied to the input $i$ terminates with a computation whose G\"odel number is $z$ and $U(z)$ is its output.

Proof To show our assertion in $WKL_{0}$ , by $\Sigma_{0}^{0}$ -comprehension, take a set $B$ of all $t\in 2^{

If we consider the corresponding assertion for general computable partial (not necessarily $\{0,1\}-$ valued) functions, it is verified in the same manner by using K\"onig’s lemma (which is equivalent to ACA over $RCA_{0}$ [$4$ , Lemma $m.7.2$]) instead of weak K\"onig’s lemma. $\overline{1RCAdenotesthesystemRCA_{0}+ful1}$(second-order) induction scheme. 95

3 Total and Partial Extension of Consistent Partial Functions

$: Next we consider compactness-like property on partial functions. For $\gamma_{1}$ \subset Xarrow Y$ and )$Q:\subset$

$dom(\gamma_{1})\subseteq dom(\gamma_{2})$ $\gamma_{1}(x)=\gamma_{2}(x)$ $Xarrow Y,$ $\gamma_{1}\preceq)9$ denotes that $\gamma_{2}$ is an extension of $\gamma_{1}$ , i.e., and for $x\in dom(\gamma_{1})$ . In general, a set $\mathscr{F}$ of partial functions is said to be consistent (cf. [2, Section 1.10]) if for all finite subset $\mathscr{G}$ of $\mathscr{F}$ , there exists $a$ (partial) function $f$ such that $\gamma\preceq f$ for each

$\gamma\in \mathscr{G}$ . As mentioned in [2, Section 1.10], one may think of a total function $f:Xarrow Y$ as being built up from tokens of information, each of which is a partial function $\sigma$ $:\subset Xarrow Y$ with finite domain. Motivated by this idea, we first define the notion of consistency over sets of finite partial functions in $RCA_{0}$ to investigate the extendability of consistent set of finite partial functions.

Definition 3. A set $\mathscr{F}$ of (codes of) finite partial functions is consistent if for all $n,$ $\gamma_{1}\in \mathscr{F}$ and 2 $\gamma_{2}\in \mathscr{F}$ such that $n\in dom(\gamma_{1})\cap dom(\gamma_{2})$ , $\gamma_{1}(n)=\gamma_{2}(n)$ holds.

On the other hand, it holds that any consistent set $\mathscr{F}$ of (not necessarily finite) partial functions has a total extension $f$ , namely, $f$ is a total function from $X$ to $Y$ such that $\gamma\preceq f$ for any $\gamma\in \mathscr{F}$ (cf. [2, Section 1.10]). Then our another goal is to investigate the strength of this assertion. Here the obstacle is that a set $\mathscr{F}$ of (not necessarily finite) partial functions is not naturally represented in the language of second-order arithmetic.3 To deal with the analogue of this assertion in second- order arithmetic, we introduce the notion of consistency over sequences of partial functions.

$i,$ $j,n\in \mathbb{N}$ Definition 4. A sequence $\langle f_{i}\rangle_{i\in N}$ of partial functions is consistent if for all such that $n\in dom(f_{i})\cap dom(f_{j})$ , $f_{i}(n)=f_{j}(n)$ holds.

We are now ready to mention our main results.

Theorem 5. The following are equivalent over $RCA_{0}.$ 1. WKL.

$a$ 2. Every consistent sequence $\langle f_{i}\rangle_{i\in \mathbb{N}}$ of partial functions has a partial extension $f$, i. e., graph $f$ ofpartialfunction such that $f_{i}\subset f$ for all $i\in \mathbb{N}.$

3. Every consistent set $\mathscr{F}$ of (codes of) finite partialfunctions has a partial extension $f$, i. e., $ a graph $f$ ofpartialfunction such that $\sigma\subset f$ for all \sigma\in \mathscr{F}.$

Proof We reason in $RCA_{0}.$ $(1arrow 2)$ : Let $\varphi(n,m):\equiv\exists i(f_{i}(n)=m)$ and $\psi(n,m):\equiv\exists i(f_{i}(n)\neq m)$ respectively. Then $\varphi$

$\Sigma_{1}^{0}$ $\Sigma_{1}^{0}$ and $\psi$ are and there is no $(n,m)$ satisfying both of $\varphi$ and $\psi$ . By separation (derived from WKL [4, Lemma IV.4.4]), we have a set $X$ such that

$(\varphi(n,m)arrow(n,m)\in X)\wedge((n,m)\in Xarrow\neg\psi(n,m))$ for all $n,m\in \mathbb{N}$ . Let

$f :=\{(n,m) : (n,m)\in X\wedge\forall m’

$2One$ can easily see that this condition is equivalent to the definition of consistency in [2, Section 1.10] over

$RCA_{0}.$ $3It$ is possible and expected to formalize this assertion in finite-type extension $RCA_{0}^{\omega}$ (cf. [3]) of $RCA_{0}$ and investigate its strength. 96

by $\Sigma_{0}^{0}$ comprehension. Then $f$ is a desired partial function.

$\mathscr{F}$ $(2arrow 3)$ : Enumerate as $\langle\sigma_{i}\rangle_{i\in N}$ and define a consistent sequence $\langle f_{i}\rangle_{i\in N}$ such that each $f_{i}$ is a partial function coded by $\sigma_{i}.$ $(3arrow 1)$ : We assume 3 to show WKL via [4, Lemma IV.4.4]. Let $g,h:\mathbb{N}arrow \mathbb{N}$ be one-to-one functions such that $g(i)\neq h(j)$ for all $i,j\in \mathbb{N}$ . By $\Sigma_{0}^{0}$ comprehension, take a sequence $\langle\sigma_{i}\rangle_{i\in \mathbb{N}}$ such that each $\sigma_{i}$ is the code of (graph of) finite partial function

$\{(2g(i),0) , (2h(i),0) , (2i+1,1$

$i$ Since $\sigma_{i}$ contains $(2i+1,1)$ , we have $i\leq\sigma_{i}$ for each (under the standard coding of finite sets

$\mathscr{F}$ $\mathscr{F}$ e.g. in [4]). Therefore, there exists the set of $\sigma_{i}$ ’s in $RCA_{0}$ . Since is trivially consistent, by our assumption 3, we have its partial extension $f$ . Taking $X:=\{n:(2n,0)\in f\}$ , it follows that

$n\in X$ $n$ $n\not\in X$ $n$ $h$ $\square$ if is in the range of $g$ and if is in the range of . This completes the proof.

Corollary 6. The following are equivalent over $RCA_{0}.$ 1. ACA.

2. Every consistent sequence $\langle f_{i}\rangle_{i\in N}$ ofpartialfunctions has a total extension $f.$

3. Every consistent set $\mathscr{F}$ offinite partialfunctions has a total extension $f.$

4. Every consistent set $\mathscr{F}$ offinite partialfunctions with single domain has a total extension $f.$

Proof. $(1arrow 2)$ follows from Theorem 5 and Theorem 1. $(2arrow 3)$ and $(3arrow 4)$ is easy. We show $(4arrow 1)$ over $RCA_{0}$ via Theorem 1. Let $f$ $:\subset \mathbb{N}arrow \mathbb{N}$ be a graph of partial function. Define $\mathscr{F}$ as the set of codes of $\{(n,m)\}$ ’s such that $(n,m)\in f$ . Then $\mathscr{F}$ is a consistent set of finite partial functions with single domain and its total extension is clearly a total extension of $f.$ $\square$

At the end, we investigate the above assertions especially for $\{0$ , 1 $\}$ -valued functions. In contrast to the previous case, all of the corresponding assertions are equivalent to weak K\"onig’s lemma over $RCA_{0}$ :

Proposition 7. Thefollowing are equivalent over $RCA_{0}.$ 1. WKL.

2. Every consistent sequence $\langle f_{i}\rangle_{i\in \mathbb{N}}$ ofpartial $\{0$ , 1 $\}$ -valued functions has a total extension $f.$

3. Every consistent sequence $\langle f_{i}\rangle_{i\in N}$ ofpartial $\{0$ , 1 $\}$ -valuedfunctions has a partial extension $f.$

$\mathscr{F}$ 4. Every consistent set offinite partial $\{0$ , 1 $\}$ -valuedfunctions has a total extension $f.$

$\mathscr{F}$ 5. Every consistent set offinite partial $\{0$ , 1 $\}$ -valuedfunctions has a partial extension $f.$ Proof We reason in $RCA_{0}.$ $(1arrow 2)$ : The idea of proof is the same as for Proposition 2. Let $B$ be the set of all $t\in 2^{

$B$ there exists an infinite path $p$ through . It is easy to see that $p$ is a desired total function. $(2arrow 3)$ , $(2arrow 4)$ , $(3arrow 5)$ , $(4arrow 5)$ are easy. $(5arrow 1)$ : The proof of $(3arrow 1)$ in Theorem 5 works as well. In fact, the consistent set $\mathscr{F}$ which we construct there consists of partial $\{0$ , 1 $\}$ -valued functions. $\square$ 97

The above proof suggests that Proposition 7 is a reverse mathematical analogue of [1, Lemma 8.17] in computability theory. Remark 8. By careful inspection, it is found that all equivalences presented in this paper can be established even over $RCA_{0}^{*}$ without the scheme of $\Sigma_{1}^{0}$ induction (See [4, Definition X.4.1] for the precise definition of $RCA_{0}^{*}$ ).

Acknowledgment

The authors are deeply grateful to their supervisor Takeshi Yamazaki for his comments and sug- gestions on the earlier version of this paper.

References

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[2] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, Second Edition, Cam- bridge University Press, 2002.

[3] U. Kohlenbach, Higher order reverse mathematics in Reverse mathematics 2001, S. Simpson ed., pp. 281-295, Lect. Notes Log. 21, Assoc. Symbol. Logic and A. K. Peters, Wellesley MA 2005. [4] S. G. Simpson, Subsystems of Second Order Arithmetic, Second Edition, Association for Symbolic Logic. Cambridge University Press, 2009.